Spin Space

We now have to discuss the wavefunctions upon which the previously introduced
spin operators act. Unlike regular wavefunctions, spin wavefunctions
do not exist in real space. Likewise, the spin angular momentum operators
cannot be represented as differential operators in real space.
Instead, we need to think of spin wavefunctions as
existing in an abstract (complex) vector space. The different members of
this space correspond to the different internal configurations of the particle
under investigation.
Note that only the directions of our vectors have any physical significance
(just as only the shape of a regular wavefunction has any physical
significance). Thus,
if the vector corresponds to a particular internal state then
corresponds to the same state, where is a complex number.
Now, we expect the internal states of our particle to be superposable,
since the superposability of states is one of the fundamental assumptions of
quantum mechanics.
It follows that the vectors making up our vector space must also be superposable.
Thus, if and are two vectors corresponding to two
different internal states then
is another vector
corresponding to the state obtained by superposing times state 1
with times state 2 (where and are complex numbers). Finally, the dimensionality of our vector
space is simply the number of linearly independent vectors required to span
it (i.e., the number of linearly independent internal states of the
particle under investigation).

We now need to define the length of our vectors. We can do this by
introducing a second, or dual, vector space whose elements are in one to one
correspondence with the elements of our first space. Let the element of the second
space which corresponds to the element of the first space
be called . Moreover, the element of the second space
which corresponds to is
. We shall assume
that it is possible to combine and in a multiplicative
fashion to generate a real
positive-definite number which we interpret as the length, or norm,
of . Let us denote this number
. Thus, we
have

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for all . We shall also assume that it is possible to combine unlike states
in an analogous multiplicative fashion to produce complex numbers. The
product of two unlike states and is denoted
.
Two states and are said to be mutually orthogonal, or independent,
if
.

Now, when a general spin operator, , operates on a general spin-state, , it coverts it into a different spin-state which we shall denote
. The dual of this state is
, where is the Hermitian conjugate of (this is the definition of an
Hermitian conjugate in spin space). An eigenstate
of corresponding to the eigenvalue satisfies

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As before, if corresponds to a physical variable then a measurement
of will result in one of its eigenvalues (see Sect. 4.10). In order to ensure that
these eigenvalues are all real, must be Hermitian: i.e., (see Sect. 4.9). We expect the to be mutually orthogonal. We
can also normalize them such that they all have unit length. In other words,

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Finally, a general spin state can be written
as a superposition of the normalized eigenstates of : i.e.,